Classifying Algebra Extensions over a fixed extension?
There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that someone out there knows about it.
I'm interested in the following situation. Let R and S be commutative rings and fix a ring homomorphism $f:R \to S$. Also fix a commutative S-algebra A. I'm interested in understanding/classifying those R-algebras B, together with a (surjective?) ring homomorphism $g: B \to A$ which intertwines the algebra structures in the following sense:
$ g(rb) = f(r) g(b)$
for all $r \in R$, and $ b \in B$. Is there a homological algebra way to do this?
A particular example that I am interested in is when we have the equality $B \otimes_R S = A$, but I am also interested in other cases as well.